Transcript Properties of Real Numbers

1-1 Properties of Real
Numbers
M11.A.1.3.1: Locate/identify irrational numbers at the approximate
location on the number line.
M11.A.1.3.2: Compare and/or order any real numbers
Objectives
Graphing and Ordering Real Numbers
Properties of Real Numbers
Number Classification
• Natural numbers are the counting numbers.
• Whole numbers are natural numbers and zero.
• Integers are whole numbers and their opposites.
• Rational numbers can be written as a fraction.
• Irrational numbers cannot be written as a fraction.
• All of these numbers are real numbers.
Number Classifications
Subsets of the Real Numbers
Q - Rational
I - Irrational
Z - Integers
W - Whole
N - Natural
Classify each number
-1
6
real, rational, integer
real, rational, integer, whole, natural
real, irrational
1
2
real, rational
0
real, rational, integer, whole
-2.222
real, rational
Properties of Real Numbers
Graph the numbers – 3 ,
4
number line.
7 , and 3.6 on a
– 34 is between –1 and 0.
Use a calculator to find that
7
2.65.
Work on Quick Check #2 on Page 6
Properties of Real Numbers
Compare –9 and –
9 = 3, so –
9. Use the symbols < and >.
9 = –3.
Since –9 < –3, it follows that
–9 < –
9.
Work on Quick Check #3 on Page 6
Properties of Real Numbers
Inverses
The Additive Inverse of any number a is -a. The sum
of opposites is 0.
The Multiplicative Inverse of any nonzero number a
is 1/a. The product of reciprocals is 1.
Properties of Real Numbers
Find the opposite and the reciprocal of each number.
1
a. –3 7
b. 4
1
Opposite: –(–3 7
Reciprocal:
1
1
–3 7
Opposite: –4
) = 3 17
=
1
– 22
=– 7
22
Reciprocal: 1
7
Work on Quick Check #4 on Page 7
4
Properties of Real Numbers
Commutative Property
• Think… commuting to work.
• Deals with ORDER. It doesn’t matter
what order you ADD or MULTIPLY.
• a+b = b+a
•4 • 6 = 6 • 4
Properties of Real Numbers
Associative Property
• Think…the people you associate
with, your group.
• Deals with grouping when you
Add or Multiply.
• Order does not change.
Properties of Real Numbers
Associative Property
•
•
(a + b) + c = a + ( b + c)
(nm)p = n(mp)
Properties of Real Numbers
Additive Identity Property
• s + 0 = s
Multiplicative Identity Property
• 1(b) = b
Properties of Real Numbers
Distributive Property
•
a(b + c) = ab + ac
•
(r + s)9 = 9r + 9s
Name the Property
•5=5+0
•
•
•
•
Additive Identity
5(2x + 7) = 10x + 35
Distributive
8•7=7•8
Commutative
24(2) = 2(24)
Commutative
(7 + 8) + 2 = 2 + (7 + 8) Commutative
Name the Property
• 7 + (8 + 2) = (7 + 8) + 2 • Associative
• 1 • v + -4 =
v + -4
• (6 - 3a)b =
6b - 3ab
• 4(a + b) =
4a + 4b
• Multiplicative
Identity
• Distributive
• Distributive
Properties of Real Numbers
The absolute value of a real number is the distance from zero on the number
line.
Simplify | 4 1 |, |–9.2|, and |3 – 8|.
3
1
1
1
1
43 is 4 3 units from 0, so | 4 3 | = 4 3 .
–9.2 is 9.2 units from 0, so |–9.2| = 9.2.
|3 – 8| = |–5| and –5 is 5 units from 0. So, |–5| = 5, and hence
|3 – 8| = 5.
Work on Quick Check #6 on Page 8
1-2 Algebraic
Expressions
M11.A.3.1.1 – Simplify/Evaluate expressions using
the order of operations to solve problems
Objectives
Evaluating Algebraic Expressions
Simplifying Algebraic Expressions
Vocabulary

A variable is a symbol, usually a letter, that
represents one or more numbers.

An algebraic expression is an expression
that contains at least one variable.

You can evaluate an algebraic expression
by replacing each variable with a value and
then applying the Order of Operations.
Order of Operations
 Parenthesis
 Exponents
 Multiply & Divide from Left to Right
 Add & Subtract from Left to Right
Example: Evaluate a(5a + 2b) if a=3 and b=-2

Substitute the values into the expression.

3[5(3) + 2(-2)]

Now apply the Order of Operations:
Inside the brackets, perform multiplication and division
before addition and subtraction

5(3) = 15 and 2(-2)= -4

3[15 + -4] then 15 + -4 = 11

3[11] = 33
Evaluating an Algebraic
Expression
Evaluate 7x – 3xy for x = –2 and y = 5.
7x – 3xy = 7(–2) – 3(–2) (5)
Substitute –2 for x and 5 for y.
= –14 – (–30)
Multiply first.
= –14 + 30
To subtract, add the opposite.
= 16
Add.
Work on Quick Check #1 on page 12
Evaluating an Algebraic
Expression with Exponents
Evaluate (k – 18)2 – 4k for k = 6.
(k – 18)2 – 4k = (6 – 18)2 – 4(6)
Substitute 6 for k.
= (–12)2 – 4(6)
Subtract within parentheses.
= 144 – 4(6)
Simplify the power.
= 144 – 24
Multiply.
= 120
Subtract.
Work on Quick Check #2 on page 12
Vocabulary
Simplifying Algebraic Expressions
A term is a number, a variable, or product of a number
and one or more variables.
The numerical factor in a term is the coefficient.
Like terms have the same variables raised to the same
powers.
Like terms: 3r 2 and  r 2
 2xy3and3xy3
Combining Like Terms
Simplify by combining like terms.
2h – 3k + 7(2h – 3k)
2h – 3k + 7(2h – 3k) = 2h – 3k + 14h – 21k
Distributive Property
= 2h + 14h – 3k – 21k
Commutative Property
= (2 + 14)h – (3 + 21)k
Distributive Property
= 16h – 24k
Work on Quick Check #4 on page 14
Finding Perimeter
Find the perimeter of this figure. Simplify the answer.
c
c
P = c + 2 + d + (d – c) + d + 2 + c + d
c
c
= c + 2+ d + d – c + d + 2 + c + d
c
c
= 2 + 2 + c + 4d
= 2c
+ c + 4d
2
= c + c + 4d
= 2c + 4d
Work on Quick Check #5 on page 14